(for Teacher2Teacher Service)
Date: Nov 22, 2003 at 11:22:10
Subject: Re: Which algorithms should we still teach in high school?
I agree with how some of the other teachers have responded, especially with
regards to the square root algorithm. I "bet" my students that they can find
the square root of a number accurate to the nearest tenth, just by using what
they already know. It's a type of mental interpolation. For example, say
you needed to know the square root of 234. You already know that 225 is 15^2
and 256 is 16^2. 234 is closer to 225 than to 256. The square root must be
closer to 15 than 16. With some practice (usually with 2 digit numbers)
students will get good at estimating. From a historical perspective,
students might be interested in a demonstration of the square root algorithm,
that is if you know it. I learned it as a 7th grader, but that was in 1970.
I find most of the teachers younger than me have never even seen it.
As to your other questions, I agree that there are a number of algorithms
that we should not spend time on, at least to the extent that we expect
students to master them. I believe, as one of the other teachers wrote, that
students learn more about math by "messing around" to find an answer or doing
an activity to increase their understanding, than just being given a set of
rules to follow. This approach takes more time, so that's why a few topics
have to go. Have you looked at your school's or state's math standards? The
NCTM Principles and Standards is another good place to go for direction.
When I choose to skip a section of a textbook, I tell students, "I've chosen
not to teach this, not because it is not important, but because there are
other things I want you to learn. I hope I am not 'marking you for life',
but know that at some future time you may be expected to learn about this."
I believe in the appropriate use of technology, but what that use is varies.
"Put a brain in their heads before you put one in their hands" is something I
heard at a workshop. At the same time, I believe that students should know
how to properly use the available technology. Things like Cramer's rule were
how mathematicians had to solve systems before technology existed to help
them. Keep in mind that logarithms were invented, at least in part, to help
the mathematicians (and astronomers) of the day deal with large numbers. No
where anymore do I see logarithms taught the way I first learned them--that
is to multiply numbers such as 2.34 x 1.56 messing with characteristics and
mantissas. What I use logarithms for and teach my students to use them for
is to solve equations using exponential functions.
As for other topics, I want my students to know about fractal geometry
(DeMoivre and powers of complex numbers apply here), about the use of
coloring theory to solve scheduling problems, mathematical methods of
apportionment, ... Once we have developed or been introduced to the
quadratic formula, I don't have my kids do it over and over by hand. I have
them write a program for their TI83 which allows them to input a, b, and c.
Writing the program and making sure it works correctly gives them a lot of
understanding about the quadratic formula.
Another thing to consider is who your audience is. My classes up to Algebra
2 are very heterogenous. A very large percentage of them won't use Algebra 2
in their lives. I'd prefer they have a good grounding in fewer topics
and have mastered those skills, as well as having "learned how to learn".
Those that go on to PreCalc and Calc are the ones who will be more
interested. They are also the ones who will "pick up" on the topics you
mentioned with a minimum of practice time.
I'd best end here. This is a topic about which I am rather passionate. Over
the years I have made many decisions about what topics to de-emphasize (or
even eliminate) and yet my students have gone on successfully, many in math
and science careers.
-Kimberley, for the T2T service
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